Thermodynamics of Sultana-Dyer Black Hole

Abstract

The thermodynamical entities on the dynamical horizon are not naturally defined like the usual static cases. Here I find the temperature, Smarr formula and the first law of thermodynamics for the Sultana-Dyer metric which is related to the Schwarzschild spacetime by a time dependent conformal factor. To find the temperature ($T$), the chiral anomaly expressions for the two dimensional spacetime are used. This shows an application of the anomaly method to study Hawking effect for a dynamical situation. Moreover, the analysis singles out one expression for temperature among two existing expressions in the literature. Interestingly, the present form satisfies the first law of thermodynamics. Also, it relates the Misner-Sharp energy ($\bar{E}$) and the horizon entropy ($\bar{S}$) by an algebraic expression $\bar{E}=2\bar{S}T$ which is the general form of the Smarr formula. This fact is similar to the usual static black hole cases in Einstein's gravity where the energy is identified as the Komar conserved quantity.